Editing Nuclear chain reaction (section)

===Effective neutron multiplication factor===

The effective neutron multiplication factor <math>k_{\mathrm{eff}}</math> is most often quantified as the ratio of the rate of neutron production to the rate of neutron loss in a nuclear system, and it is often described using the [[six factor formula|six-factor formula]]. 

<math>k_{\mathrm{eff}} = {\mbox{rate of neutron production} \over \mbox{rate of neutron loss}}</math>

Using <math>k_{\mathrm{eff}}</math> and the prompt neutron lifetime, <math>l</math>, the following differential equation can be used to describe the time rate of change of the neutron population:

<math>{ d \over dt } n(t) = \left({ k_{\mathrm{eff}} - 1 \over l }\right) n(t)</math>

When solved for <math>n(t)</math>, this equation represents the neutron population <math>n</math> at any given time <math>t</math> given an initial neutron population <math>n(0)</math> at <math>t=0</math>:

<math>n(t) = n(0)e^{\left( { k_{\mathrm{eff}} - 1 \over l } \right) t}</math>

When describing a nuclear reactor, where neutron population is directly proportional to thermal power, the following equation is used:

<math>P = P_0 e^{t / \tau} </math>

where <math>P</math> is the reactor power at time <math>t</math>, given an initial power <math>P_0</math>, and <math>\tau</math>, the [[reactor period]]. The value of <math>\tau</math> can be calculated as

<math>\tau = { l \over k_{\mathrm{eff}}-1 }</math>

==== Six-factor formula ====
The effective neutron multiplication factor <math>k_{\mathrm{eff}}</math> can be described using the product of six probability factors that describe a nuclear system. These factors, traditionally arranged chronologically with regards to the life of a neutron in a [[Thermal-neutron reactor|thermal reactor]], include the probability of fast non-leakage <math>P_{\mathrm{FNL}}</math>, the fast fission factor <math>\varepsilon</math>, the resonance escape probability <math>p</math>, the probability of thermal non-leakage <math>P_{\mathrm{TNL}}</math>, the thermal utilization factor <math>f</math>, and the neutron reproduction factor <math>\eta</math> (also called the neutron efficiency factor). The six-factor formula is traditionally written as follows:

<math>k_{\mathrm{eff}} = P_{\mathrm{FNL}} \varepsilon p P_{\mathrm{TNL}} f \eta</math>

Where:

* <math>P_{\mathrm{FNL}}</math> describes the probability that a [[Neutron temperature|fast neutron]] will not escape the system without interacting. 
** The bounds of this factor are 0 and 1, with a value of 1 describing a system for which fast neutrons will never escape without interacting, i.e. an infinite system.
** Also written as <math>L_f</math>
* <math>\varepsilon</math> is the ratio of total fissions to fissions caused only by thermal neutrons
** Fast neutrons have a small probability to cause fissions in uranium, specifically uranium-238.
** The fast fission factor describes the contribution of fast fissions to the effective neutron multiplication factor
** The bounds of this factor are 1 and infinity, with a value of 1 describing a system for which only thermal neutrons are causing fissions. A value of 2 would denote a system in which thermal and fast neutrons are causing equal amounts of fissions.
* <math>p</math> is the ratio of the number of neutrons that begin thermalization to the number of neutrons that reach thermal energies.
** Many isotopes have "resonances" in their capture [[Neutron cross section|cross-section]] curves that occur in energies between fast and thermal.
** If a neutron begins thermalization (i.e. begins to slow down), there is a possibility it will be absorbed by a non-multiplying material before it reaches thermal energy.
** The bounds of this factor are 0 and 1, with a value of 1 describing a system for which all fast neutrons that do not leak out and do not cause fast fissions eventually reach thermal energies.
* <math>P_{\mathrm{TNL}}</math> describes the probability that a thermal neutron will not escape the system without interacting.
** The bounds of this factor are 0 and 1, with a value of 1 describing a system for which thermal neutrons will never escape without interacting, i.e. an infinite system.
** Also written as <math>L_{th}</math>
* <math>f</math> is the ratio of number of thermal neutrons absorbed in by [[Fissile material|fissile]] nuclei versus the number of neutrons absorbed in all materials in the system.
** This factor describes the efficiency of thermal neutron utilization in the system, hence the name thermal utilization factor.
** The bounds of this factor are 0 and 1, with a value of 1 describing a system for which the entire system is made of fissile nuclei (i.e. thermal neutrons can only react with fissile materials). Similarly, a value of 0.5 describes a system for which reactions with fissile and non-fissile nuclei are equal.
** For a conventional nuclear power reactor, this factor is the only one that can be directly controlled by the operator. With manipulations to the [[control rod]]s, you can increase the amount of neutrons being absorbed in non-fissile nuclei while simultaneously decreasing the amount of neutrons absorbed in fissile nuclei.
* <math>\eta</math> describes the probability that a neutron absorbed will cause a fission reaction.
** This factor describes the behavior of the fissile material, specifically if a neutron is absorbed, how likely is it to cause a fission, and how many neutrons does the fission produce.

The multiplication factor is sometimes calculated with a simplified [[four factor formula|four-factor formula]], which is the same as described above with <math>P_{\mathrm{FNL}}</math> and <math>P_{\mathrm{TNL}}</math> both equal to 1, and is used when an assumption is made that the reactor is "infinite" in that neutrons are very unlikely to leak out of the system. This value <math>k_\infty</math> is often used in safety evaluations of reactor designs.

==== Criticality ====
Because the value of <math>k_{\mathrm{eff}}</math> is directly related to the time rate of change of the neutron population in a system, it is convenient to classify the state of the nuclear system with regards to the critical value of the neutron population equation. The point at which the behavior of a nuclear system shifts is when <math>k_{\mathrm{eff}}</math> is exactly equal to 1. This point is called "criticality," and describes a system in which the production rate and loss rate of neutrons is exactly equal.

When <math>k_{\mathrm{eff}}</math> is less than or greater than one, the terms subcriticality and supercriticality are used respectively to describe the system:
* <math>k_{\mathrm{eff}} < 1</math> ([[subcriticality]]): The neutron population in the system is decreasing exponentially. 
* <math>k_{\mathrm{eff}} = 1</math>  ([[critical mass|criticality]]): The neutron population is maintaining a constant value.
* '''<math>k_{\mathrm{eff}} > 1</math>'''  ([[supercriticality]]): The neutron population in the system is increasing exponentially.
In a practical nuclear system, like a fission reactor, if criticality is intended it is likely that <math>k_{\mathrm{eff}}</math> will actually oscillate from slightly less than 1 to slightly more than 1, primarily due to thermal feedback effects. The neutron population, when averaged over time, appears constant, leaving the average value of <math>k_{\mathrm{eff}}</math> at around 1 during a constant power run. Both delayed neutrons and the transient fission product "[[burnable poison]]s" play an important role in the timing of these oscillations.

==== Reactivity ====
The value of <math>k_{\mathrm{eff}}</math> is generally not easy to calculate or use practically. Instead, a system's [[Reactivity of a nuclear reactor|reactivity]] is quantified instead. The reactivity of a nuclear system is qualitatively described as the departure from criticality. The equation below describes the pure reactivity <math>\rho</math> as a function of the neutron multiplication factor <math>k_{\mathrm{eff}}</math>:

<math>\rho = {k_{\mathrm{eff}} - 1 \over k_{\mathrm{eff}}}</math>

or when comparing the reactivity differences between two nuclear systems with multiplication factors <math>k_1</math> and <math>k_2</math>,

<math>{ \Delta k \over k } = {k_2 - k_1 \over k_1 k_2}</math>

For most systems, the reactivity <math>\rho</math> has a very small range, making any value difficult to qualitatively describe or interpret, like <math>k_{\mathrm{eff}}</math>. Often, it is expressed in units of <math>%\Delta k/k</math>, [[per cent mille]], or (almost solely in the United States) with the derived units of [[Dollar (reactivity)|dollars and cents]]. Note that <math>\rho</math> is often also expressed as <math>\Delta k / k</math>

<math>% { \Delta k \over k} = \rho \times 100</math>

<math>\mathrm{pcm} = \rho \times 10^5</math>

<math>$ = {\rho \over \beta_{\mathrm{eff}}}</math>

The value <math>\beta_{\mathrm{eff}}</math> is known as the effective delayed neutron fraction, and it describes the fractional contribution of delayed neutrons to the fission rate of the system and is quantified as the ratio of the total number of fissions caused by delayed neutrons to the total number of fissions in a system. This number is slightly different than the delayed neutron fraction <math>\beta</math>, which is the fraction of neutrons in the system that are [[Delayed neutron|delayed]], because delayed neutrons are generally born at lower energies, and thus are easier to thermalize, meaning they are more likely to cause a fission than a prompt neutron. This weighting effect is given in the derivation of <math>\beta_{\mathrm{eff}}</math>.

==== Subcritical multiplication ====
When a nuclear system is subcritical, an introduction of neutrons to the system will result in that population decaying away; however, if neutrons are introduced at a constant rate (i.e. from a neutron source), a nuclear system can appear critical while not actually maintaining true criticality. This is called '''source criticality''' and due to a phenomenon called subcritical multiplication.

The neutron population equation can be modified to be written as follows:

<math>{d \over dt}\left(n(t)\right) = \left({{k_\mathrm{eff}-1} \over l}\right)n(t)+S(t)</math>

This is a much more difficult differential equation to solve. In this case, we assume that all neutrons are from the source, and that each generation of neutrons is of equal magnitude. In this case, we can approximate using a geometric series:

<math>n(t) = n_0 + n_0 k_\mathrm{eff} + n_0 \left(k_\mathrm{eff}\right)^2 + n_0 \left(k_\mathrm{eff}\right)^3 + \cdots = n_0 \sum_{i=0}^\infty (k_\mathrm{eff})^i = \left({1 \over {1-k_\mathrm{eff}}}\right) n_0</math>
[[File:Graphical Representation of Subcritical Multiplication.jpg|alt=A bar graph showing a series of colored bars, with one color each representing an exponentially decreasing generation of neutrons. The bars stack on each other, such that as each generation decreases, the total population in the next time frame is larger, resulting in the total behavior is increasing, approaching a stable value.|thumb|350x350px|Graphical representation of subcritical multiplication]]
We take the above equation and define a new factor <math>M</math>, called the subcritical multiplication factor:

<math>M = {1 \over {1-k_{\mathrm{eff}}}}</math>

Multiplying this factor by the source strength (in neutrons/sec) will give the stable neutron population, as long as <math>k_{\mathrm{eff}}</math> is known:

<math>n_\infty = S_0 \times M</math>

Much more commonly, this equation is used to estimate <math>k_{\mathrm{eff}}</math>, as the stable neutron population is easy to measure, but it is difficult to know the strength of a neutron source. To get around this, as a system approaches criticality, <math>M</math> approaches infinity; therefore, it is much more practical to measure <math>1/M</math>, which approaches zero as a system approaches criticality. <math>1/M</math> can be approximated by the ratio of count rates before and after a reactivity addition. 

<math>{CR_0 \over CR} \approx {1 \over M} \quad \therefore \quad \lim_{ CR_0/CR \rightarrow 0 } \left({k_\mathrm{eff}}\right) = 1</math>

Most neutron sources are a combination of an alpha particle emitter and beryllium. [[Beryllium-9]], the only naturally-occurring stable isotope of beryllium, is capable of emitting a neutron when an alpha particle is absorbed. This (<math>\alpha, n</math>) binary reaction is what generates neutrons. The most common of these are americium-beryllium (AmBe), plutonium-beryllium (PuBe), and polonium-beryllium (PoBe) sources.

<chem>{^9_4 Be} + {^4_2 \alpha} \Rightarrow {^1_0 n} + {^{12}_{6} C}</chem>

[[Antimony-124]] is also used in conjunction with beryllium to generate neutrons, as the gamma ray emitted by antimony-124 is at a unique energy that can be absorbed by beryllium and cause it to emit a neutron. This is called a (<math>\gamma, n</math>) reaction. Antimony-124 sources are commonly used to locate beryllium ore by mining companies. 

<chem>{^9_4 Be} + {\gamma} \Rightarrow {^1_0 n} + {^{8}_{4} Be}</chem>

Other sources of neutrons are from accelerators that use fusion to generate neutrons using deuterium and tritium fusion via this reaction

<chem>{^2_1 D} + {^2_1 D} \Rightarrow {^1_0 n} + {^3_2 He}</chem>

<chem>{^2_1 D} + {^3_1 T} \Rightarrow {^1_0 n} + {^4_2 He}</chem>
==== Special reactivity cases ====
Not all neutrons are emitted as a direct product of fission; some are instead due to the [[radioactive decay]] of some of the fission fragments. The neutrons that occur directly from fission are called "prompt neutrons", and the ones that are a result of radioactive decay of fission fragments are called "delayed neutrons". The fraction of neutrons that are delayed is called <math>\beta</math>, as discussed before, and this fraction is typically less than 1% of all the neutrons in the chain reaction.<ref name="Duderstadt" />

As the delayed neutron precursors (the radionuclides that decay via neutron emission) have decay constants on the order of seconds and milliseconds, the delayed neutrons born from them allow the neutron population in a system to respond to small reactivity changes several orders of magnitude more slowly than just prompt neutrons would alone, as these delayed neutrons effectively increase the mean neutron lifetime <math>l</math>.<ref name="Lamarsh" /> Without delayed neutrons, changes in reaction rates in nuclear systems would occur at speeds that are too fast for humans to control.

When <math>\beta_{\mathrm{eff}}>0</math> and <math>\rho = 0</math>, a nuclear system is called '''[[Delayed criticality|delayed critical.]]''' The region of supercriticality where <math>0 < \rho < \beta_{\mathrm{eff}}</math> is known as '''delayed supercriticality.''' It is in this region that all nuclear power reactors operate. When <math>\rho = \beta_{\mathrm{eff}}</math>, the system is described as '''prompt critical'''. The region of supercriticality for <math>\rho > \beta_ {\mathrm{eff}}</math> is known as '''prompt supercriticality.''' This is the region in which nuclear weapons operate, alongside some pulsing nuclear research reactors, like the [[TRIGA]] reactor.
[[File:Criticality Diagram.png|alt=A graph with a horizontal line denoting the point at which k_effective equals 1 and five sets of bars composed of two sections stacked on each other labeled "prompt" (bottom) and "delayed" (top). The five categories moving left to right are "Subcritical," "Delayed Critical," "Delayed Supercritical," "Prompt Critical," and "Prompt Supercritical." The bars are lengthening from left to right. For subcriticality the bar is below the line. For delayed critical, the bar is just touching the line. For delayed supercritical, the delayed section of the bar is above the line. For prompt critical, the delayed section is completely above the line and the prompt section is just touching the bar. For prompt supercritical, the delayed section is completely above the line, and the prompt section extends partially above the line.|center|thumb|400x400px|Graphical representation of delayed and prompt criticality and supercriticality]]